Minimal surfaces and Schwarz lemma

Abstract

We prove a sharp Schwarz lemma type inequality for the Weierstrass–Enneper parameterization of minimal disks. It states the following. If $F:D\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma \subset R^3$, where $D$ is the unit disk and $\left|\Sigma \right|=\pi R^2$, then $\left|F_x\left(z\right)\right|(1-{\left|z\right|}^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine image of a disk, and $F$ is linear up to a Möbius transformation of the unit disk.